1997/11/03 by Jan Stienstra, Stienstra, Jan · 4 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons #alg-geom #hep-th #math.AG
paper · pdf · doi:10.48550/arxiv.alg-geom/9711002
37 pages Latex2e; one picture; submitted for publication in the proceedings of the Taniguchi Symposium 1997 "Integrable Systems and Algebraic Geometry"
arxiv created 1997/11/03 · openalex publication_date 1997/11/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they give solutions for certain resonant systems of GKZ hypergeometric differential equations. For this some complex parameters in the Gamma-series are replaced by nilpotent elements from a ring RA,T. The adapted Gamma-series is a function Ψ with values in the finite dimensional vector space RA,T⊗ C. Applications of these results in the context of toric Mirror Symmetry are described. Building on work of Batyrev we show that the relative cohomology module of a certain hypersurface in a torus is a GKZ hypergeometric D-module which over an appropriate domain is isomorphic to the trivial D-module RA,T⊗ OT, where OT is the sheaf of holomorphic functions on this domain. The isomorphism is explicitly given by adapted Gamma-series. As a result one finds the periods of a holomorphic differential form of degree d on a d-dimensional Calabi-Yau manifold, needed for the B-model side input to Mirror Symmetry. Relating our work with that of Batyrev and Borisov we interpret the ring \cR\sA,\gT as the cohomology ring of a toric variety and a certain principal ideal in it as a subring of the Chow ring of a Calabi-Yau complete intersection. This interpretation takes place on the A-model side of Mirror Symmetry.